Bublz! : Playing with Bubbles to
Develop Mathematical Thinking
Dhruv Chand
Mudit Sinha
National Institute of Technology Vellore Institute of Technology
Karnataka
VIT University
Surathkal, Karnataka, India
Chennai, Tamil Nadu, India
[email protected]
[email protected]
Karthik Gopalakrishnan
Shreya Sriram
Indian Institute of Technology
Delhi Technological University
Patna
New Delhi, India
Patna, Bihar, India
[email protected]
Abstract
We encounter mathematical problems in various forms
in our lives, thus making mathematical thinking an
important human ability [4]. In this paper, we present
Bublz!, a simple, click-driven game for children to
engage in and develop mathematical thinking in an
enjoyable manner.
Author Keywords
[email protected]
Game Design; Mathematics; Reasoning; Mathematical
Thinking; Strategic Thinking;
Nisha K K
ACM Classification Keywords
National Institute of Technology
K.3.1;
Karnataka
Figure 1: Bublz! Start Screen
Surathkal, Karnataka, India
[email protected]
Author names are arranged in alphabetical order by last name. Copyright
is held by the authors.
Introduction
A complete understanding of mathematics not only
includes knowledge of mathematical concepts and
principles, but also the capacity to engage in
mathematical thinking: observing patterns, examining
constraints, making conjectures and inferences, and so
on [3]. Mathematics is a dynamic process of “gathering,
discovering and creating knowledge in the course of
some activity having a purpose” [2]. Such activities or
tasks typically have more than one solution strategy
and involve decision-making and interpreting the
reasonableness of possible actions [5].
Figure 2: Instructions Screen
Bublz! (Figure 1) is a single-player game in which
players are given a task of this nature in each game
level. The players need to do simple arithmetic and
effectively reason with numbers, which are depicted
using bubbles, in order to complete the task. The only
mathematical prerequisite to play the game is the
ability to do addition and subtraction, which are
depicted through the creation and removal/popping of
bubbles respectively. We developed a fully-functioning
digital prototype for the task with six levels of play
using Unity 3D [6] in the JavaScript and C#
programming languages. The game is primarily
intended to be played on a computer, but it can very
easily be modified and built to work on portable devices
like phones and tablets as well.
Game Play
For each of the six levels in the game, a unique triplet
of positive numbers (L, D, R) is associated with the
three common types of clicks done on a mouse: single
left-click, double left-click and single right-click (Figure
2).
Figure 3: Main game screen at the start
of a level, showing one bubble to start
making moves on
When a level starts, one bubble is displayed on the
screen (Figure 3). A positive target number T randomly
chosen from the integers between 2 and 70, both
inclusive, and the triplet associated with the three
types of clicks for the level, are also displayed on the
screen. The player is given an initial score of 1000.

When the player does a single left-click on a
bubble, L new bubbles are created.

When the player does a double left-click on a
bubble, D new bubbles are created.

When the player does a single right-click on a
bubble, R bubbles are removed or “popped”,
including the bubble on which the click was done.
The player's task is to obtain T bubbles on the screen
using a combination of the three types of clicks, which
are the possible types of moves. To obtain the highest
possible score of 1000, the player should do it in the
least number of moves in which T bubbles can be
obtained on the screen. With each move made beyond
the least number of moves, the player's score reduces
by 10.
Feedback on performance (Figure 4) is provided to
players upon completing a level to facilitate learning
[1]: the number of moves in which the player obtained
T bubbles and the minimum number of moves in which
the player could have obtained it are displayed on the
screen. The player can retry the same target, repeat
the level with a different target, or try targets in the
next level. A video trailer of the game is available at
the following link: vimeo.com/132009026
Game Viewed as an Optimization Problem
The task for a player in each level is to minimize the
total number of moves made in order to maximize the
score – an optimization problem. Let (L, D, R) be the
triplet for a given level. Let T be the displayed target.
Let the number of single left-clicks, double left-clicks
and single right-clicks done to create T bubbles be x, y
and z respectively. In order to obtain the highest score
in a given level, the total number of moves must be
minimized, hence the formulation of the optimization
problem is:
Minimize N = x + y + z, such that
1 + Lx + Dy – Rz = T
serves as a hint to the player that the target can be
reached without ever exceeding 150 bubbles.
x, y, z are non-negative integers
Evaluation
Game Design
For each level, the triplet values (L, D, R) are chosen
such that any positive number can be generated using
a linear combination of L, D and –R. For example, (3, 4,
1) is a valid triplet because any positive number can be
generated using a linear combination of 3, 4 and -1.
This ensures that there is always at least one sequence
of moves to generate T bubbles.
Figure 4: Score screen after finishing a
level, showing the score, number of moves
made, optimal number of moves, and
various game options
Figure 5: Invalid move screen is
displayed when an attempted move would
cause the number of bubbles to go outside
the range 1-150
Intuition suggests more bubbles should be generated
on doing a double left-click since the effort put in is
twice the effort put in to do a single left-click. Hence,
with the aim of making the game easy to grasp, L and
D are chosen such that D > L.
Triplet values of higher magnitudes are chosen for
higher levels to make them more difficult. For example,
if the current level has (3, 4, 1) as the click triplet, the
next level could have (5, 7, 4) as the click triplet.
To ensure that there will always be at least one bubble
on the screen that can be clicked, the game prevents
the player from doing a single right-click move when
the number of bubbles on the screen is lesser than or
equal to the value R. This is done by displaying an
appropriate message (Figure 5). To ensure that the
player doesn’t unnecessarily overload the screen with
bubbles by doing single/double left-clicks all the time,
the game also prevents the player from making a
single/double left-click move if making that move would
result in the total number of bubbles on the screen
exceeding a threshold value of 150. This additionally
Our immediate goal after developing the game was to
get feedback from children and evaluate our design
choices. Hence, we organized a preliminary play
session for 10 middle school students from a local
school in Bangalore, India, and observed while they
played. The play session involved playing three
successive levels of the game.
We made several observations from the play session.
As the play session progressed, most students who got
large targets almost instantly did a double left-click as
their first move. This suggests that in the course of
playing the game, the students developed a mental
representation associating the number of left-clicks
with the number of bubbles they could generate, a
positive feedback to our design decision of generating
more bubbles for a double left-click than for a single
left-click. The students also generally took greater time
to complete higher levels and additionally reported
during post-play feedback interviews that they found
higher levels challenging, a positive feedback to our
decision to use triplets of higher magnitudes in higher
levels. No student overloaded the screen with bubbles,
making the upper cap of 150 on the number of bubbles
seem unnecessary. We observed the time students
spent playing the game and determined their intention
to replay the game through post-play interviews.
According to both factors, students showed clear
motivation to play the game. They reported liking and
enjoying the game. Some students wanted the game to
be more visually appealing while others wanted more
levels to play, which we took into account to improve
the game. When asked about the game’s utility to
them, most students felt the game would make them
“better at mathematics”.
Carnegie Mellon’s Internship Program in Technology
Supported Education, which brought us together for 15
days to conceptualize, develop and conduct a
preliminary evaluation of this game.
Future Work
We plan to make refinements to the game, such as
using achievement stars instead of numeric scores, a
popular mechanic in many games at present, and also
removing the count of bubbles present on the screen to
make it more challenging. We also plan to undertake a
formal study of the impact that regular play of the
game has on mathematical thinking in children. One
way is to study the correlation that known measures of
mathematical thinking have with an edit-distance like
metric that tells how close a child’s performance is at
any point of time to that of an ideal player using ingame performance data. It would also be interesting to
see whether regularly playing the game leads to
improved performance on optimization problems
encountered in later stages of life, such as active stock
trading with the goal of maximizing profit.
Acknowledgements
We thank Amy Ogan, Erin Walker and Erik Harpstead
for their guidance and assistance in developing Bublz!.
We thank MS Ramaiah Institute of Technology,
Bangalore, India, for enabling us to play-test our game
with children. We also thank Carolyn P. Rosé and
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